 # more pixels are better!

Started Dec 14, 2008 | Discussions thread
Re: Are those figures for "print" or "screen"? (nt)

Les Olson wrote:

ejmartin wrote:

How pedantic do you want me to be? I do not mean "has image data at
higher spatial frequencies if there are higher spatial frequencies in
the image"; there is always data up to the Nyquist frequency in an
image. The amplitude of the signal at high spatial frequency can be
small if there is no fine scale detail. Such is the case for a
region of uniform tonality.

Well, pedantic enough to remember that the whole and sole basis of
saying that SD of pixel output is "noise" is that noise is the only
cause of variation of pixel output.

You cannot call SD of pixel output "noise" when there is variation
among pixels due to signal variation (ie, image modulation). You
cannot calculate pixel SD from an image with both (photon + read +
dark) noise and signal variation and label that "noise".

I'm glad to see we can set aside trivial and obvious statements.

Of course there are high frequency components in noise: the
distribution of the pixel outputs only approximates to Normal,
however much the sample size increases. And as sample size gets
smaller the distribution gets rougher, but it would be absurd to call
that statistical noise sensor noise.
--

The issue is not how well the histogram of a uniform patch approximates the normal distribution, as the size of the sample is varied, though of course that can also be calculated on the basis of the statistics of the noise. And it is not the high frequency components of noise that are most responsible for this variation, it is the low frequency components (those of frequency below the inverse of the patch size in appropriate units).

Rather, the point is that spatially random data has a power spectrum of fluctuations of uniform magnitude in frequency space (in an ensemble average). When averaged over orientations, that leads to a noise power N(k) linear in the magnitude of the spatial frequency, N(k)=c k for some constant 'c' (essentially the covariance of the noise spectrum, times 2 pi). Ignoring some inessential caveats, the std dev is the square root of the area under this linear curve. The square root of the area under the curve is proportional to the Nyquist frequency and that is why finer pixel pitch cameras show higher noise standard deviation even if they have the same value of the constant 'c' and hence the same noise power spectrum at low frequency as a coarser pixel pitch camera. I gave the example of the 40D/50D as an illustration of this fact.

Now, for a given signal S, uniform over an image region, the S/N ratio is not a single number, it is the function of spatial frequency S/N(k). What is usually quoted as the S/N is the signal S divided by the standard deviation of the noise. Again, the latter will be lower for finer pixel pitch cameras, since it is inversely proportional to the Nyquist frequency, even if S/N(k) is the same function as a coarser pixel pitch camera being compared.

What is of greater relevance to the visual appearance of the image is S/N(k) at a fixed, reference k and not S/(std dev), especially when the Nyquist frequency is well beyond the resolution of vision at a given output size and viewing distance. Roughly, S/(std dev) is closely related to pixel peeping levels of SNR, and S/N(k) at a reference k is closely related to what DxO calls "normalized for print" levels of SNR. Though of course N(k) contains much more information, as I illustrated with the example of grain size.

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emil
-- http://theory.uchicago.edu/~ejm/pix/20d/

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